How can I achieve this type of spherical unwrapping?
The idea is to deform a sphere and report its UVs on another sphere.
Use an icosphere with some subdivision (say 6).
In edit mode, change its tri to quads: select all and AltJ (this matters for the UV tranfer operation later).
Back in object mode, copy this sphere.
Enter edit mode, in front view, select the four faces at the center. Turn proportional editing on, scale and enlarge the proportional editing so that its around all the vertices.
You should have something like this:

Once done, in front view, edit mode, select all and unwrap U then &quot;project from view (bounds)&quot;.
Back to object mode, select the original, shift select the scaled sphere and make links CtrlL and choose &quot;transfer UV map&quot;.

And, voilà:

Other side:

Scaled UVs: